Given n general points p 1 , p 2 , . . . , p n ∈ P r , it is natural to ask when there exists a curve C ⊂ P r , of degree d and genus g, passing through p 1 , p 2 , . . . , p n . In this paper, we give a complete answer to this question for curves C with nonspecial hyperplane section. This result is a consequence of our main theorem, which states that the normal bundle N C of a general nonspecial curve of degree d and genus g in P r (with d ≥ g + r) has the property of interpolation (i.e. that for a general effective divisor D of any degree on C, either H 0 (N C (−D)) = 0 or H 1 (N C (−D)) = 0), with exactly three exceptions.

Given an abelian variety A of dimension g over a number field K, and a prime ℓ, the ℓ n -torsion points of A give rise to a representation ρ A,ℓ n : Gal(K/K) → GL 2g (Z/ℓ n Z). In particular, we get a mod-ℓ representation ρ A,ℓ : Gal(K/K) → GL 2g (F ℓ ) and an ℓ-adic representation ρ A,ℓ ∞ : Gal(K/K) → GL 2g (Z ℓ ). In this paper, we describe the possible determinants of subquotients of these two representations. These two lists turn out to be remarkably similar.Applying our results in dimension g = 1, we recover a generalized version of a theorem of Momose on isogeny characters of elliptic curves over number fields, and obtain, conditionally on the Generalized Riemann Hypothesis, a generalization of Mazur's bound on rational isogenies of prime degree to number fields.The study of associated characters of these kinds goes back to Serre's foundational work on the Open Image Theorem, which states that for an elliptic curve E without complex multiplication, the action of the absolute Galois group of a number field on the adèlic Tate module. This is proved in two principal steps. First, in [16], Serre shows that the ℓ-adic image ρ E,ℓ ∞ : G K → GL 2 (Z ℓ ) has finite index for all ℓ. Second, in [17], Serre shows that for sufficiently large primes, the mod-ℓ image ρ E,ℓ : G K → GL 2 (Z ℓ ) is surjective. In each case, the proof consists of reducing the problem to the study of the ℓ-adic and mod-ℓ associated characters of E respectively, i.e. studying the Galois action on the 1-dimensional subquotients.A major conjecture, which is still open, is whether the index of the image of G K in GL 2 ( Z) in Serre's theorem is bounded uniformly in E. The first step towards this result, for K = Q, is Mazur's seminal theorem on isogenies [9]. This theorem is equivalent to the statement that, for an elliptic curve E over Q and for a prime ℓ > 163, the ℓ-torsion module E[ℓ] is irreducible (equivalently, no isogenies E → E ′ defined over Q have kernel of order ℓ). An essential step of Mazur's proof is to analyze the possible associated characters (up to torsion of small degree) of subquotients of E[ℓ], and show that for ℓ > 163 the list of possible associated characters is empty.Momose in [13] gives an exhaustion (i.e. a list containing all possibilities, perhaps with excess) for the mod-ℓ associated characters of elliptic curves over number fields K attached to subquotients of E[ℓ], for ℓ sufficiently large depending on K. When K is quadratic, either real, or imaginary of class number greater than one (i.e. as long as K = Q[ √ D] for D ∈ {−1, −2, −3, −7, −11, −19, −43, −67, −163}), the list of possible associated characters is empty. In particular, any elliptic curve E over such a quadratic field K has irreducible torsion module E[ℓ] (equivalently, admits no ℓ-isogenies) as long as ℓ > C K for some constant C K that depends only on K.The main theorem of our paper gives an analogous exhaustion for abelian varieties of dimension g over K. When applied to elliptic curves, we obtain slightly stronger versions of the above results of Mom...

Let C be a general curve of genus g, embedded in P r via a general linear series of degree d. In this paper, we prove the Maximal Rank Conjecture, which determines the Hilbert function of C ⊂ P r . DegenerationsIn this section, we outline the degenerations we shall use in the proof of Theorem 1.3.

Given an elliptic curve E over a number field K, the ℓ‐torsion points E[ℓ] of E define a Galois representation Gal(K¯/K)→GL2(픽ℓ). A famous theorem of Serre (Invent. Math. 15 (1972) 259–331) states that as long as E has no complex multiplication (CM), the map Gal(K¯/K)→GL2(픽ℓ) is surjective for all but finitely many ℓ. We say that a prime number ℓ is exceptional (relative to the pair (E, K)) if this map is not surjective. Here, we give a new bound on the largest exceptional prime, as well as on the product of all exceptional primes of E. We show in particular that conditionally on the generalized Riemann hypothesis, the largest exceptional prime of an elliptic curve E without CM is no larger than a constant (depending on K) times log NE, where NE is the absolute value of the norm of the conductor. This answers affirmatively a question of Serre (Inst. Hautes Études Sci. Publ. Math. (1981) 323–401).

In a recent paper, Bruinier and Ono prove that the coefficients of certain weight −1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. In particular, for the partition function p(n), they prove thatwhere P is a weak Maass form and α Q ranges over a finite set of discriminant −24n + 1 CM points. Moreover, they show that 6 · (24n − 1) · P (α Q ) is always an algebraic integer, and they conjecture that (24n − 1) · P (α Q ) is always an algebraic integer. Here we prove a general theorem which implies this conjecture as a corollary.